It really depends what you mean here. If you mean can those born physically female, those who identify and live as women, and those who are socially coded as women have their own spaces then the answer is yes, in fact that's is what has happened for a hundred years or more. There is growing evidence of trans people living in stealth in Victorian times and beyond, it's near certain they had have used single sex spaces (as much as they existed) inline with their aquired gender - to do anything else would have been far too dangerous. So this is the system we have had for pretty much as long as single sex spaces have existed. It is a system which runs on pretty basic common sense, with a few minor legal safeguards such as the single sex exemption, and which has shown itself to be workable and safe over the decades. It may not be perfect, masculine appearing women sometimes get harassed, some people might feel nervous around a masculine appearing woman or trans woman, and things like single sex toilets can cause problems for parents and people with carers, but it's what we've had for a long time and there doesn't seem any real social appetite outside of a few quarters to change that.
If you mean can women (or men) have their own spaces in which they can guarantee that every single persons who uses it is XX/XY or has the correct genitals then that is probably socially impossible without horrifyingly draconion measures. Firstly which do you want, chromosones, genitals or both? And how do you check? Which of the constellation of biological sex indicators and social signifiers should be required to enter a single sex space without harassment? Could we no longer have male doctors, porters and visitors in women's hospital wards in case someone was made to feel uncomfortable by the presence of a male? Should male healthcare workers, guards and maintenance staff be kept out of women's prisons? Should every single sex space have the same assesssment criteria you might have in a refuge, and even now that assessment cannot be fully guaranteed. And if this is the case, and we're all carrying sex ID cards, then why stop there? Why not keep out people with certain criminal records? Or from certain socially marginalised groups? What kind of society would this look like, what other social attitudes might and pressures might emerge around gender and presentation? How would this impact on trans, intersex or gender nonconforming people? Perhaps you'd need to ban crossdressing, just in case, it's been illegal in societies before and still is in a few countries. Perhaps there would be calls for segregation to be further extended to other spheres.
The extreme conservative and religious right is pushing this so hard they are prepared to drop every principle they possess and work with lesbians and pro-choice feminists to achieve it. Why is that? If these aims were ever brought to fruition would they be implemented by a radical feminist government or something very different? Is this really the society people want? The reason there is a split in feminism over this is not because younger feminists have been brainwashed by the woke into 'being kind', or even just that there is a social shift towards trans acceptance but because some have the imagination to see where some of these demands might lead.